First of all, starting from the Wigner-Kirkwood expansion [1]} of Bloch density function, one derives TF equation of state just as an 0th order approximation which works only for very slow varying densities, therefore, supplementary higher order (gradient) corrections could be added. This is done trough a Weiszacker term \(\lambda (\nabla ^2\rho ^{1/2})/\rho ^{1/2}\)  with \(\lambda =1/8\)  [2]} which should account, in principle, for local oscillations of the density profile [3]}.
